1
Frequency Regulation in Hybrid Power Systems using Particle Swarm Optimization and Linear Matrix Inequalities based Robust Controller
Design
Shashi Kant Pandey a, Soumya R. Mohanty a, Nand Kishor a, João P. S. Catalão b,c,d*
a Motilal Nehru National Institute of Technology, Allahabad 211004, India b University of Beira Interior, R. Fonte do Lameiro, 6201-001 Covilha, Portugal
c INESC-ID, R. Alves Redol, 9, 1000-029 Lisbon, Portugal d IST, University of Lisbon, Av. Rovisco Pais, 1, 1049-001 Lisbon, Portugal
Received 22 March 2014; received in revised form 25 June 2014 Abstract It is often necessary to investigate the output power against load demand in a system having distributed generation (DG) resources connected to
the existing conventional power system. In this paper, the load frequency control (LFC) problem is presented using different optimization
algorithms for two types of power system configurations: (i) hybrid configuration of thermal power system (TPS) integrated with DG,
comprising wind turbine generators (WTGs), diesel engine generators (DEGs), fuel cells (FCs), aqua-electrolyzer (AE) and battery energy
storage system (BESS); (ii) two area interconnected power system with DG connected in area-1. The inclusion of wind energy system in DG,
having high variability in its output power, results into a challenging task for the realization of an effective controller design. This difficulty is
further enhanced with random variation of load demand. The control scheme proposed in this paper is based on linear matrix inequalities (LMI)
with its parameters tuned by particle swarm optimization (PSO), as a new contribution to earlier studies. The robustness of this controller is
thoroughly demonstrated in the above hybrid power systems with different conditions of load disturbances, wind power and parameter variations.
Keywords: Distributed generation, load frequency control, robust control, stabilization, time delay.
* Corresponding author at: University of Beira Interior, R. Fonte do Lameiro, 6201-001 Covilha, Portugal. Tel.: +351 275 329914; fax: +351 275 329972.
E-mail address: [email protected] (J.P.S. Catalão).
2
Nomenclature f System frequency deviation (p.u.Hz)
WTGP Change in wind turbine power generation (p.u.MW)
WPP Change in available wind power (p.u.MW)
WTGK Gain constant of the WTG
WTGT Time constant of the WTG (sec)
AEP Change in aqua electrolyzer power (p.u.MW)
AEK Gain constant of the AE
AET Time constant of the AE (sec)
FCP Change in FC power generation (p.u.MW)
FCK Gain constant of the FC
FCT Time constant of the FC (sec)
DEGP Change in diesel power generation (p.u.MW)
DEGK Gain constant of the diesel generator
DEGT Time constant of the diesel generator (sec)
BESSP Change in BESS power generation (p.u.MW)
BESSK Gain constant of BESS
BESST Time constant of the BESS (sec)
gK Governor gain constant
gT Governor time constant (sec)
tT Steam turbine constant
rK Reheat gain constant
rT Reheat time constant of the steam turbine R Drooping characteristic (Hz/p.u.MW)
pK Power system gain (Hz/p.u.MW)
pT Power system time constant (sec)
EX Small adjustment in position of governor valve (p.u.MW)
tP Small adjustment in thermal turbine thermal output (p.u.MW)
1. Introduction
The installation of distributed generation (DG) resources has being increased to meet the growing energy demand. In general, a
DG system makes use of small electric power generation resources located nearby to its consumers and load centers. These
generation resources include, among others, wind energy, diesel generator, fuel cells and energy storage systems. To meet the
increased load demand of an isolated community, expansion of DG systems may be achieved through interconnection with
conventional generation resources. The resulting hybrid power system intends to provide reliable and high quality service to its
consumers, and this in turn depends mostly on the type and action of the controller implemented in the system.
Integration of DG resources especially based on wind turbines imposes new challenge to power systems control, making the
electric power industry become more complicated.
3
In such hybrid system, deviations in load demand and stochastic variation in wind power adversely affect the frequency, so it is
necessary to preserve the power balance between generation and demand, being achieved through automatic load frequency
control (LFC) in some acceptable range. Compliance with frequency regulation policies set by regulation authorities becomes
imperative. The frequency control issue in power systems having high penetration of wind systems is addressed in [1]-[3]. It is
important to investigate the impact of high wind power penetration with conventional power flow in the overall area tie-line
power. The wind system output power fluctuation dynamics has negative contribution to the power imbalance and thus to the
frequency deviation, which should be taken into account in the existing LFC control scheme.
The control scheme implemented seems to be ineffective against wide the range of uncertainties in operation. In fact, frequency
deviation in significant range may lead to under/over frequency relay trip and thus disconnection of system loads and generation.
The study in this paper is related to the frequency regulation issue in hybrid power systems, with DG resources having negative
impact on system frequency profile.
The different methods to tune gains of PI/PID controllers are given in [4]-[5]. A Linear Matrix Inequalities (LMI) based linear
quadratic regulator (LQR) control design for a wind farm is proposed in [6]. The design of robust PID controller using LMI
approach is described in [7]. The application of H∞ approach in controller design has played an important role in the study of
numerous system dynamics since its original formulation. An integration of LMI and static output feedback represents a new
approach in the design of PI/PID controllers. The LFC problem using LMI approach with consideration of time delay is also
presented by some authors [8]-[11]. In [8] the authors described the LFC with communication delays using LMI approach. Bevrani
et al. [9] proposed a decentralized PI control approach with communication delays being considered as model uncertainty. Design
of decentralized robust PI-based LFC for time-delay system is discussed in [10]. Frequency regulation for time-delay power
system using LMI approach is described in [11].
The method for PI control design which uses a mixture of H∞ control and genetic algorithm (GA) method to tune the gains of PI
is proposed in [12]. Robust analysis and controller design for LFC is given in [13] and decentralized LFC for multi area power
system is given in [14].
In the past, researchers have reported studies on LFC in a conventional system that consists of thermal/hydro or a combination
of them using several variants of optimization techniques to design controller gains [15]-[20]. The performance of such design
approach, however, depends not only on the optimization techniques, but also on the objective function. The authors of [18] have
used bacteria foraging optimization technique for designing a frequency controller.
The scheme of LFC based on multi-objective GAs is described in [21]. The design of a robust decentralized controller for LFC
of multi-area interconnected power systems is discussed in [22]-[23].
4
It is worth to mention that in the study considered here, an isolated DG system with conventional power system operates in a
local region and is not wide spread over a large geographical region.
Thus, it becomes imperative to perform a study on the hybrid DG system. Further, the uncertainty of intermittent renewable
energy resources with generation fluctuation may result in unintentional structure changes, which will further exaggerate the
challenge for stabilizing the frequency response.
Recently, some authors [24]-[29] have reported their study analysing the influence of energy storage in frequency deviation
considering renewable resources based DG system. A hybrid power system in an island for frequency control is described in [24].
Small-signal analysis of a hybrid renewable system with energy storage is discussed in [25]-[27]. The robust H∞ LFC in hybrid
systems is discussed in [28]-[29].
This paper explores, as a new contribution to earlier studies, the use of particle swarm optimization (PSO) and LMI - PSOLMI
based controller design - to achieve minimum frequency deviation. The simulation results are demonstrated on two types of power
system configurations. The first one is a hybrid configuration of thermal power system (TPS) integrated with DG, which
comprises wind turbine generators (WTGs), diesel engine generators (DEGs), fuel cells (FCs), aqua-electrolyzer (AE) and battery
energy storage system (BESS). The second configuration is a two area interconnected power system comprising of DG resources
in area-1 with communication delay. The study investigates frequency deviation profile caused by a sudden change in generation
and load demand in these two configurations. The robust control scheme via PSOLMI is tested under all possible disturbances,
including the intermittent nature of wind speed and other uncertainties.
This paper is organized as follows; the optimization algorithms are given in Section II, followed by the design strategy for two
system configurations in Sections III & IV, respectively. The simulation results are given in Sections V & VI, and lastly the
conclusions are provided in Section VII.
2. Controller Design Algorithms
This section describes an outline of various control designs via LMI approach.
2.1 H∞ Control Design via LMI Approach (H∞)
The objective of H∞ control theory is to design the control law u on the basis of measured variable y , so that the effect of
disturbance w on control variable z , given in terms of infinity norm of the transfer function from z to w , z wT does not
surpass a specified limit demarcated as assuring robust performance. The classical closed-loop system through H∞ robust control
is represented as Fig. 1, in which P(s) represents a linear-invariant system and K∞(s) represents robust H∞ controller [12].
5
( )P s
( )K s
w
u y
z
Fig. 1. Closed-loop system via robust H∞ control
State space representation of system model is given by:
1 2
1 1 1 1 2
2 1 2 2 2
x A x B w B uz C x D w D uy C x D w D u
(1)
where x is the state variable, w is the disturbance and other exterior input vector, u is the control input, z and y are controlled
and measured output vectors, respectively.
The state space for controller is considered as:
k k
k k
A B yu C D y
(2)
where is the state vector for controller.
From (1) and (2), the subsequent closed loop state-space model is obtained:
cl cl cl cl
cl cl cl
x A x B wz C x D w
(3)
where
c lx
x
2 2 2
2
k kc l
k k
A B C C B CA
B C A
1 2 2 2
2 2
kc l
k
B B C DB
B D
1 1 2 2 1 2c l k kC C D D C D C 11 12 22cl kD D D D D
Closed-loop root-mean-square (RMS) gain ( )T s or H∞ norm of transfer function z wT
does not surpass performance index
, if and only if there is a symmetric matrix X such that:
2
0
T Tc l c l c l c l
T Tc l c l
c l c l
A X X A B X CB I D
C X D I
(4)
0X (5)
Therefore, the optimal H∞ control is attained by the minimization of performance index , subject to matrix inequalities (4) and (5).
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2.2 H∞ Control Design for Time Delay Systems
Consider a time-delay system in the subsequent form:
1 2
1
2
( ) ( ) ( ) ( ) ( ) ( )( ) ( )
( ) ( ), ( ) ( ) [ max( , ), 0]
d hx t Ax t A x t d B w t B u t B u t hz t C x t
y t C x t x t t t d h
(6)
where h and d denote the delay quantities in input and state, respectively. ( )t is a vector-valued continuous initial function.
Theorem 1 (given below) adjusts H∞ model in control synthesis for time-delay systems and creates the conditions for state
feedback control law ( ) ( )u t K t stabilization of (6) and to ensure the H∞ norm bound of closed-loop transfer function z wT
,
namely ; 0.z wT
Theorem 1- State feedback controller K asymptotically studies time-delay system (6) ; , 0z wT d h
and if there occurs
symmetric matrices 1 20, 0, 0P Q Q fulfilling the LMI
1 2 1 1
1
2
12
1
0 0 000 0 0
0 0 00 0 0
T T T T Tc c d h
d
hT
P A A P Q Q A P K B P C B PP A Q
P B K QC IP B I
where 2cA A B K .
The LFC problem time-delayed is condensed to the creation of static output feedback (SOF) control ( ( ) ( ))u t ky t for time-
delay system provided in (6), using Theorem 2. From control law 2( ) ( ) ( )u t ky t kC x t , and considering Theorem 1, there occurs
a memory less feedback control with a constant gain 2K kC . Therefore, 2 2cA A B kC .
Theorem 2- The SOF control asymptotically stabilizes system (6) and ; , 0z wT d h
if there occurs symmetric matrices
0, 0, 0a bY Q Q satisfying the following LMI:
2 2 2 1 1
2 2
2
12
1
( ) ( )0 0 0 0 0
0 0 0 0 000 0 0 0 0
0 0 0 0 00 0 0 0 00 0 0 0 0
T T T T T Ta b d h
d a
h bT
A Y Y A Q Q B k C Y Y A B k C Y C Y BB k C I
Y IW A Y Q
B k C Y QY C I
B I
In order to define the SOF controller k , the next minimization problem has to be solved:
m in0 , 0 , 0 , 0 .
, , , a ba b
su b jec t to Y Q Q WQ Q Y k
The proof of Theorem 2 is given in [11].
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2.3 LMI representation of LQR (LMI-LQR)
Consider a linear time-invariant approach for the proposed power system-1 with the following state space model.
x A x B u dy C x
u K y
(7)
( )x A BK C x d (8)
The variables of the proposed hybrid power system-1 are TE t r WP AE FC DEG BESS DGSx X P P f P P P P P P
And TDGSy f P . The details of LMI representation of LQR are described in [6].
2.4 Control Design using GALMI Approach
The GA is nowadays a popular optimization methodology in many application fields, mostly due to their robust properties in
attaining the optimum solution and the capability of reaching a nearly optimal one close to the global minimum solution.
The search mechanism in GA is based on natural selection and persistence of the fittest. The objective function is:
min z wT (9)
where z wT is infinity norm of transfer function from z to w that does not exceed a specified limit .
The robust control design procedure for LFC using GALMI technique is hereafter provided. Consider the state space
representation of the system given by:
1 2
1 12
2
x A x B w B uz C x D u
y C x
(10)
The control signal is:
u Ky (11)
The chosen controller corresponds to a static output feedback controller, and has inferior complexity than typical H∞ control
design shown in (2). To define the control parameter vector K , from (10) and (11) it can be obtained (12)-(13):
2u KC x (12)
c l c l cl c l
c l cl cl
x A x B wz C x D w
(13)
where 2 2clA A B KC , 1clB B , 1 12 2clC C D KC 0clD .
The parameters of the controller specified by K are found with the assistance of GAs for minimizing performance
index given by (9) determined by the LMIs constraints (4) and (5).
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2.5 Control Design using PSOLMI Approach
Particle swarm optimization (PSO) algorithms are population-based evolutionary search techniques with several advantages:
they are simple, fast and easy to code. They also require small storage space. PSO approaches are advantageous over GAs. GA is a
metaheuristics algorithm based on population size, selection, mutation and crossover, whereas in PSO, with proper choice of
parameters and expression of velocity and displacement, the optimization can be more effectively achieved. For instance, each
particle in PSO keeps the best local solution as well as the best global solution. The initial population in PSO remains constant and
time consuming operators do not exist. If ( )ix t indicates the position of the ith particle in search space at time step t, the new
position of the particle is found by adding the velocity component ( )iv t :
( 1) ( ) ( 1)i i ix t x t v t (14)
In general, PSO algorithms are grouped in two general categories: the best global solution (gbest), and the best local solution
(lbest). In this paper, the gbest algorithm is employed. In gbest PSO algorithm, the vicinity of each particle involves the whole
group. In this case, updating the social component of the particle’s velocity involves the information acquired from entire particles
in the group.
The social information ˆ( )y t is the best position determined by the swarm. For gbest PSO, particle i velocity is computed by:
1 1 2 2 ˆ( 1) ( ) ( )[ ( ) ( )] ( )[ ( ) ( )]ij ij j ij ij j j ijv t v t c r t y t x t c r t y t x t (15)
where ( )ijv t indicates the velocity of particle i at dimension 1,2,..., xj n at time step t, ( )ijx t is the position of particle i at
dimension j, 1c and 2c are acceleration factors and 1 ( )jr t and 2 ( )jr t are random variables with normal distribution in [0,1].The last
two variables incorporate randomness in the algorithm.
The proposed PSOLMI design uses PSO procedure to find the optimum values for PI controller, considering the H∞ constraints
in terms of LMIs in (4) and (5). This brings about the robust performance of H∞ control for PI controller and makes system
vulnerable against bounded uncertainties. The problem formulation for PSOLMI is the same as GALMI, as discussed previously.
3. Case Study-1: Hybrid Power System
The LFC study being limited to small perturbations, the system models are usually adequately represented by a linearized one.
The interconnected conventional reheated thermal source along with DG resources forming a hybrid power system is provided in
Fig. 2.
9
refP
dP1R
(a) Block diagram
1g
g
KsT
11
r r
r
s K Ts T
11 tsT 1
P
P
KsT
1R
WTGP1
WTG
WTG
KsT
1AE
AE
KsT
1FC
FC
KsT
1DEG
DEG
KsT 1
BESS
BESS
KsT
LP
f
2u
1u
DGSP
EX tP rP
WPP
AEP
FCP
DEGP
BESSP
eP
C o n tro ller
C o n tro ller
(b) Transfer function model
Fig. 2. Case study-1: Hybrid power system configuration
The DG system under study consists of energy resources such as WTG, FC, AE, DEG, and BESS. The dynamic behavior of a
power system in the presence of wind power generation systems might be different from conventional power plants. The power
outputs of such sources are depending on weather conditions seasons and geographical location. When wind power is a part of the
power system, additional imbalance is created when the actual wind power deviates from its forecast due to wind speed variations.
So, scheduling conventional generators units to follow load (based on the forecasts) may also be affected by wind power. Because
wind power is an intermittent source which differs from the traditional plants, the control of wind turbine output is much more
challenging. When demand is beyond the limit of available wind power, stability problems may arise. Therefore, an appropriate
coordination between stability and controllability of active power in wind turbines should be further defined. A part of wind power
generator output is utilized by AE for hydrogen production, which is used in FC to generate power. The BESS is used in the power
system to perform the function of load levelling. The mathematical models of the different subsystems in the DG power system are
presented as follows [25]-[27]:
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Wind Power Turbine
The power of the wind turbine generator depends upon the wind speed. Wind speed varies continuously with time. The
mechanical power output of the wind turbine is proportional to the cube of wind speed. The wind power is given as:
31
2W P R P WP A C V (16)
where : the air density (kg/m3); :RA the swept area of blade (m2) and :PC Power coefficient, a function of tip speed ratio ( )
and blade pitch angle ( ) , :WV wind speed. The wind turbine system has several nonlinearities. When wind turbine uses its pitch
controller to counteract utility grid frequency oscillations, its output power varies. Hence, the pitch angle set point is a nonlinearity
limited by the boundaries of variation in output power. The pitch system, which is used to turn the pitch angle according to wind
speed, introduces nonlinearity. The transfer function of wind turbine generator is given by first-order lag neglecting non-linearities:
( )
1WTG WTG
WTGWP WTG
P KG s
P sT
(17)
Aqua-Electrolyzer for Hydrogen Production
A part of the wind power generator output is utilized by aqua electrolyzer for hydrogen production, which is used in the fuel cell
to generate power. The transfer function of the AE can be expressed by first-order lag as:
2
( )1
AE AEAE
AE
P KG s
u sT
(18)
Fuel Cell Power Generation
The FC is an electrochemical device, which converts the chemical energy of fuel (hydrogen) into electrical energy by
combining gaseous hydrogen with air in the absence of combustion. They are considered to be an important resource in hybrid
distributed power systems due to several advantages, such as high efficiency and low pollution. FC generator is a higher order
model and has non linearity. For low frequency domain analysis, it is represented by a first order lag transfer function as:
2
( )1
FC FCFC
FC
P KG s
u sT
(19)
Diesel Engine Power Generation
The DEG is in action autonomously to supply the deficit power in the hybrid DG system to meet the supply-load demand
balance condition. A diesel generator is a nonlinear system because of the presence of a nonlinear, time-varying dead time
between the injection and production of mechanical torque. The transfer function of DEG can be given by linear first order lag as:
2
( )1
DEG DEGDEG
DEG
P KG s
u sT
(20)
11
Battery Energy Storage System
The BESS is used to provide additional damping to power system swings to improve both transient and dynamic stability.
The wind power has an intermittent nature, and short time power fluctuation may cause large problems for power systems
operation. A possible solution is use of energy storage devices.
The BESS is very affective to store large amount of wind power due to its very good technical characteristics, like large energy
density and fast access time. The transfer function of the BESS can be first order lag given as:
2
( )1
BESS BESSBESS
BESS
P KG s
u sT
(21)
The power-frequency balance is achieved by control signals 1u and 2u fed from the designed controller. The details of
mathematical models of different modules in the hybrid DG power system are described in [25]-[27], as well as the data of
different components in the DG power system under study.
Power balance is attained by the following equation:
e D G S T H LP P P P (22)
where eP is the error in power supply, D G SP is the output power of DG system, ( )T H rP P is the output power of
reheat thermal system and LP is the change in load demand.
The total output power of the hybrid DG system is expressed by:
D G S W P F C A E D E G B E S SP P P P P P (23)
where , , , ,W P F C A E D E G B E S SP P P P P are the power generation by the WTG, FC, AE, DEG and BESS, respectively. The
impact of wind power variation on system frequency response is a key factor to analyse the LFC issue in hybrid systems
comprising DG resources.
The deviation in the frequency outline f is given by e
sys
PfK
, where s y sK is the system frequency characteristic constant.
The transfer function for system frequency variation to p.u. deviation in power is given by:
1(1 ) 1
Psys
e sys sys P
KfTP K sT sT
(24)
where PK and PT are the equivalent gain constant and time constant, respectively.
12
In steady state condition, the power generation is matched with load demand, driving the frequency deviation to zero. The
balance between power generation and load demand is achieved by virtue of change in frequency to generate the control signal
through the controller, as shown in Fig. 2(b). The input exogenous signals 1w and 2w are the load disturbance and wind power
variation, respectively, i.e.
1
2
L
WTG
w Pand
w P
(25)
Now, consider the state space model as (10) with following state:
T
E t r WP AE FC DEG BESS DGSx X P P f P P P P P P f
1 2Tw w w , 1 2
Tu u u , T
y f f
Robust controller K is capable of minimizing the fictitious output z in the occurrence of w . Thus, the vector z should
adequately cover all signals that are to be minimized to achieve the desired response. Therefore, a suitable fictitious output vector
may be chosen as:
1 2 3 1 4 2
Tz f f u u
(26)
where i ( 1,2.....4)i are constraint weighting coefficients and should be selected by the designer to achieve the preferred
performance.
The control signal for the proposed power system-1 is written as:
1 11
2 22
p ip i
p i
fK Kuu K f K f Ky
K K fu
(27)
The controller sends a command signal to the conventional power system (thermal) and DEGs, FCs, AE and BESS in order to
regulate the active power and thereby regulate the system frequency. In order to define the control parameter vector K , the closed-
loop system is obtained as (13).
4. Case Study-2: Two-Area Power System with DG
The conventional framework of the power system is modified with the incorporation of thermal and DG in area-1 and thermal
power in area-2, respectively. The block diagram of the proposed structure is shown in Fig. 3(a) and its transfer function model is
presented in Fig. 3(b).
13
(a) Block diagram
1
11g
g
KsT
1 1
1
11
r r
r
sK TsT
1
11 tsT
1
11p
p
KsT
WTGP1
WTG
WTG
KsT
1AE
AE
KsT
1FC
FC
KsT
1DEG
DEG
KsT 1
BESS
BESS
KsT
1LP
1f
3u
1u
DGSP
1EX 1tP 1rP
WPP
AEP
FCP
DEGP
BESSP
eP
PI
2
21g
g
KsT
2 2
2
11
r r
r
sK TsT
2
11 tsT
2
21p
p
KsT
2LP
2f2u
2EX 2tP 2rPeP
122 Ts
12a
2
1R2
1R
2
1R 2
1R
12a
1ACE
2ACE
TIEP
1Area
2Area
Thermal
D G
T h erm al
d se hse
PI hse
PId se hse
(b) Transfer function
Fig. 3.Case study-2: Two area power system with DG
The LFC model in traditional structure with time delays is discussed in [8] and [10]. In interconnected 2-area power systems the
communication delay comes into play, which affects the controller performance. In view of this, in the control input and output in
each area of the LFC structure, the communication delays are depicted as shown in Fig. 3(b). An exponential function se is used
to describe the communication delay, where represents the communication delay time. To generate the area control error (ACE)
signal, the detected frequency and tie-line power deviations via communication line is used.
14
The plant state space model is developed as:
1 1 1 1 2 2 2 2 1 2
T
E t r E t r TIE WP AE FC DEG BESSx X P P f X P P f P P P P P P ACE ACE 1 1 2 2
Ty A C E A C E A C E A C E
1 1T
L L WTGw P P P
1 2 3Tu u u u
1 1 2 2 3 4 1 5 2 6 3T
TIEz f f P u u u
where i ( 1,2.....6)i are constraint weighting coefficients and should be chosen by the designer to achieve the wanted
performance. The ACE signal in multi-area LFC structure is given by:
i i i TIEiAC E f P (28)
where TIEiP is the net exchange of tie-line power of control ith area. The control signal is written as:
1
1 1 11
2 2 22
3 3 3
2
0 00 0
0 0
p i
p i p i
p i
ACEu K K ACE
u K ACE K ACE u K K kyACE
u K KACE
(29)
5. Results and Discussion on Case Study-1
This section discusses the simulation analysis with the above described controllers in hybrid DG power system. The LFC scheme
in an interconnected hybrid power system should control the area frequency as well as the tie-line power flow between the control
areas.
o Load demand change
The controller response is tested for step load and random load change.
Step load change: The load disturbance and wind power variation applied to the hybrid DG system are considered as
0.01LP pu and 0.01WTGP pu . The frequency deviation profile ( f ) for various controllers is presented in Fig. 4. With the
PSOLMI, the frequency deviation of the system is quickly damped with comparatively reduced undershoot. The frequency
deviation takes about 10 sec to settle down to the steady-state.
Fig. 4. Response with step load change
0 5 10 15 20 25 30-4
-2
0
2x 10-5
Time [Sec]
Freq
uenc
y de
viat
ion
[pu]
GALMILMILQRPSOLMIH-INFINITY
15
Random step load change: The load disturbance variation in steps as shown in Fig. 5(a) with 0.01WTGP pu is applied to the
system. The system frequency deviation is provided in Fig. 5(b). It is observed that PSOLMI controller achieves comparatively
better damping for frequency deviation profile. The control effort of the proposed PSOLMI is improved comparatively to the other
controllers in terms of low overshoot/undershoot and faster settling time.
(a) Load change pattern (b) Frequency deviation
Fig. 5. Response with random step load change
Random load change: The random load variation as shown in Fig. 6(a) and 0.01WTGP pu is applied to the system. System
response to frequency deviation is shown in Fig. 6(b). The PSOLMI controller performance remains consistent as discussed
previously. The response is characterized by low overshoot/undershoot, less oscillation and faster response.
(a) Random load variation (b) Frequency deviation
Fig. 6. Response with random load variation
0 10 20 30 40-0.06
-0.04
-0.02
0
0.02
0.04
Time [sec]
Ran
dom
load
var
iatio
n [p
u]
0 10 20 30 40-10
-5
0
5x 10-6
Time [Sec]
Freq
uenc
y de
viat
ion
[pu]
GALMILMILQRPSOLMI
0 5 10 15 20 25 30-0.01
-0.005
0
0.005
0.01
Time [Sec]
Ran
dom
load
var
iatio
n [p
u]
0 5 10 15 20 25 30-5
0
5x 10-6
Time [Sec]
Freq
uenc
y de
viat
ion
[pu]
PSOLMILMILQRGALMI
16
Both GA and PSO are similar in the sense that these two techniques are population-based search methods and they search for
the optimal solution by updating generations. Since the two approaches are supposed to find a solution to a given objective
function but employ different strategies and computational effort, it is appropriate to compare their performance. It should be
noted that while the GA is inherently discrete, i.e. it encodes the design variables into bits of 0’s and 1’s, therefore it easily handles
discrete design variables, PSO is inherently continuous and must be modified to handle discrete design variables. Compared with
GA, PSO has some attractive characteristics. It has memory, so knowledge of good solutions is retained by all particles; whereas
in GA, previous knowledge of the problem is destroyed once the population changes. PSO has constructive cooperation between
particles, so particles in the swarm share information between them.
The response with PSOLMI is much faster, with less overshoot and settling time compared to GALMI and LMILQR. The
PSOLMI controller has good damping characteristics to low frequency oscillations and stabilizes the system much faster. This
extends the power system stability limit and the power transfer capability.
o Input wind power change
Step change in wind power input: The random step change in wind power input as shown in Fig. 7(a) is applied to the system
with 0.01LP pu . The variation in system frequency deviation is presented in Fig. 7(b). The response achieved by PSOLMI
controller is again better (low overshoot/undershoot and fast settling time) than other controllers.
(a) Step change in wind power (b) Frequency deviation
Fig. 7. Response for case study-1 with step change in wind power
Random wind power input: The random wind power input as shown in Fig. 8(a) with 0.01LP pu is applied to the system.
Fig. 8(b) shows the corresponding system frequency deviation. With the proposed PSOLMI controller, the oscillation of frequency
deviation is significantly diminished, suggesting robust performance of the proposed controller against stochastic variations of
wind power due to weather conditions as input disturbance.
0 10 20 30 400.01
0.015
0.02
0.025
0.03
Time [Sec]
Ran
dom
win
d po
wer
[pu]
0 10 20 30 40-1
0
1
2x 10-5
Time [Sec]
Freq
uenc
y de
viat
ion
[pu]
LMILQRPSOLMIGALMI
17
(a) Random wind power variation (b) Frequency deviation
Fig. 8. Response with random wind power variation
o Parameter variations
Any controller designed for fixed plant parameter or tuned to nominal conditions may not perform satisfactory with changes in
their nominal values. Also, in some cases, the value of these parameters is not accurately determined, either experimentally or
mathematically. So, a controller design is desired to be robust enough against uncertainties in parameter values of the system
model. The robustness of the controllers is evaluated by computation of a quantitative performance index, the integral square error
(ISE) index given as:
302
0
( )ISE f d t (30)
The computed values of ISE for the various controllers are depicted in Fig. 9. As observed for the change in parameters in the
range of +30% to -30% from nominal value, the corresponding value of index is computed to be higher for LMI-LQR and
GALMI, compared to PSOLMI controller. Also, it is evident that a consistent value of the index is obtained. These results
successfully validate that the proposed controller not only compensates against parameters variations but also regulates the
frequency deviation profile adequately. It is found that robustness is guaranteed under both upper bound (+30%) and lower bound
(-30%) of parameter changes.
Fig. 9. Computed ISE for uncertainties in all system parameters
0 5 10 15 20 25 300
0.005
0.01
0.015
0.02
Time [Sec]
Ran
dom
win
d po
wer
[pu]
0 5 10 15 20 25 30-6
-4
-2
0
2
4x 10-6
Time [Sec]
Freq
uenc
y de
viat
ion
[pu]
PSOLMILMILQRGALMI
-30% -20% -10% 0% +10% +20% +30%0
1
2
3
4x 10-6
Parameter variation
Inte
gral
squ
are
erro
r [pu
]
PSOLMIGALMILMILQR
18
6. Results and Discussion on Case Study-2
This section discusses the simulation analysis conducted on case study-2 using different optimization algorithms. It becomes
imperative to investigate the response variation, when existing thermal with DG resources (area 1) are interconnected to another
thermal system (area 2).
6.1 Load demand change
The response of the controllers is tested for step load and random load change.
Step load change: The load disturbance and wind power variation applied to the power system-2 is considered as
1 0.01LP pu , 2 0.01LP pu and 0.01WTGP pu . The responses obtained are shown in Fig. 10.
Fig. 10(a) and (b) represent the frequency deviation in areas 1 and 2, respectively. The frequency deviation takes about 1.0sec to
reach steady-state. The tie-line power deviation is shown in Fig. 10(c). Fig. 10(d) and (e) represent the ACE in area-1 and area-2,
respectively. By using the proposed method, PSOLMI, the frequency deviation of the system is quickly damped with
comparatively reduced undershoot.
The time-domain characteristics (undershoot and settling time of frequency deviation) using PSOLMI controller is lower than
those obtained by GALMI controller. The peak frequency deviation remains low and has reduced significantly in shorter time
period in case of PSOLMI controller. Fig. 10(f), (g) and (h) represent the control signals. The control effort required by PSOLMI
controller is low and also for short time span.
Random step load change: Consider the load disturbance being varied in steps as shown in Fig. 5(a) in area-1 with
2 0.01LP pu and 0.01WTGP pu applied to the system. The system responses are illustrated in Fig.11 (a-c). It is observed that
PSOLMI controller achieves comparatively better damping for frequency deviation profile and optimal tie-line power changes.
Thus, the obtained results illustrate better performance despite load disturbances. Further, comparison of Fig. 5 (case study-1) and
Fig. 11 (case study-2), a smaller settling time, but an increased overshoot/undershoot in the dynamics of frequency deviation is
observed in latter system under the same load disturbance. This is due to the fact that, during the load disturbance, the tie line
supports the control area by absorbing power.
Random load variation: The random load variation as shown in Fig. 6(a) in area-1 with 2 0.01LP pu and 0.01WTGP pu is
considered. The frequencies and tie-line power deviations are shown in Fig. 12(a-c). The PSOLMI controller performance
suggests a general trend, i.e. remains consistent as discussed previously. The response is characterized by low
overshoot/undershoot, less oscillation and faster response. The control scheme provides smooth performance under transient
characteristic of disturbances. On the other hand, as compared to Fig. 6(b), the frequency deviation dynamics is accompanied by
higher amplitudes and oscillation about the steady state value.
19
(a) Frequency deviation in area-1 (b) Frequency deviation in area-2
(c) Tie-line power deviation (d) ACE in area-1
(e) ACE in area-2 (f) Control signal u1
(g) Control signal u2 (h) Control signal u3
Fig. 10. Response for case study-2 with step load change
0 1 2 3-6
-4
-2
0
2x 10-3
Time [Sec]
Freq
uenc
y de
viat
ion
F1 [p
u]
GALMIPSOLMI
0 0.5 1 1.5 2 2.5 3-3
-2
-1
0
1x 10-3
Time [Sec]
Freq
uenc
y de
viat
ion
F2 [p
u]
GALMIPSOLMI
0 1 2 3-2
-1
0
1x 10-3
Time [Sec]
Tie-
line
pow
er [p
u]
GALMIPSOLMI
0 1 2 3-4
-2
0
2x 10-3
Time [Sec]A
CE
-1 [p
u]
GALMIPSOLMI
0 1 2 3-2
0
2x 10-4
Time [Sec]
AC
E-2
[pu]
GALMIPSOLMI
0 0.5 1 1.5 2-2
-1
0
1x 10-3
Time [Sec]
Con
trol s
igna
l u1
[pu]
GALMIPSOLMI
0 1 2 3-5
0
5
10
15x 10-5
Time [Sec]
Con
trol s
igna
l u2
[pu]
GALMIPSOLMI
0 1 2 3-0.3
-0.2
-0.1
0
0.1
Time [Sec]
Con
trol s
igna
l u3
[pu]
PSOLMIGALMI
20
(a) Frequency deviation in area-1 (b) Frequency deviation in area-2
(c) Tie-line power deviation
Fig. 11. Response for case study-2 with random step load change
(a) Frequency deviation in area-1 (b) Frequency deviation in area-2
(c) Tie-line power deviation
Fig. 12.Response for case study-2 with random load variation
0 10 20 30 40-0.4
-0.2
0
0.2
0.4
Time [Sec]
Freq
uenc
y de
viat
ion
F1 [p
u]
PSOLMIGALMI
0 10 20 30 40-0.2
-0.1
0
0.1
0.2
Time [Sec]
Freq
uenc
y de
viat
ion
F2 [p
u]
PSOLMIGALMI
0 10 20 30 40-0.1
-0.05
0
0.05
0.1
Time [Sec]
Tie-
line
pow
er [p
u]
PSOLMIGALMI
0 5 10 15 20 25 30-0.1
-0.05
0
0.05
0.1
Time [Sec]
Freq
uenc
y de
viat
ion
F1 [p
u]
PSOLMIGALMI
0 5 10 15 20 25 30-0.04
-0.02
0
0.02
0.04
Time [Sec]
Freq
uenc
y de
viat
ion
F2 [p
u]
PSOLMIGALMI
0 5 10 15 20 25 30-0.02
-0.01
0
0.01
0.02
Time [Sec]
Tie-
line
pow
er [p
u]
PSOLMIGALMI
21
6.2 Input wind power change
Consider the load disturbance, 1 0.01LP pu , 2 0.01LP pu and random wind power input applied to the system. The
frequency and tie-line power deviation profiles are shown in Fig. 13. The proposed controller PSOLMI performs satisfactorily,
minimizing the frequency deviation in area 1, having strong interactions with area 2. These results indicate improved performance
accommodating the wind power uncertainty. A close observation suggests comparatively higher frequency deviation in area 1 with
respect to area 2.
(a) Frequency deviation in area -1 (b) Frequency deviation in area -2
(c) Tie-line power deviation
Fig. 13. Response for case study-2 with random wind power variation
The change in wind power generation, change in AE power generation, change in FC power generation, change in diesel power
generation, change in BESS power generation, change in total DG power generation, change in power generation in area-1 and
change in power generation in area-1 for load disturbance and wind power variation of 1 0.01LP pu , 2 0.01LP pu , and
0.01WTGP pu are all shown in Fig. 14 (a-h), respectively.
0 5 10 15 20 25 30-0.01
-0.005
0
0.005
0.01
Time [Sec]
Freq
uenc
y de
viat
ion
F1 [p
u]
GALMIPSOLMI
0 5 10 15 20 25 30-3
-2
-1
0
1
2
3x 10-3
Time [Sec]Fr
eque
ncy
devi
atio
n F2
[pu]
GALMIPSOLMI
0 5 10 15 20 25 30-2
-1
0
1
2x 10-3
Time [Sec]
Tie-
line
pow
er [p
u]
GALMIPSOLMI
22
(a) Change in wind power generation (b) Change in AE power generation
(c) Change in FC power generation (d) Change in diesel power generation
(e) Change in BESS power generation (f) Change in total DG power generation
(g) Change in power generation in area-1 (h) Change in power generation in area-2
Fig. 14. Response of change in power generation for case study-2 with step load change
0 5 10 15 20 25 300
1
2
3
4
5
6x 10-3
Time [Sec]
Cha
nge
in w
ind
pow
er g
ener
atio
n [p
u]
GALMIPSOLMI
0 5 10 15 20 25 30-0.12
-0.1
-0.08
-0.06
-0.04
-0.02
0
0.02
Time [Sec]
Cha
nge
in A
E p
ower
gen
erat
ion
[pu]
PSOLMIGALMI
0 10 20 300
0.5
1
1.5
2
2.5
3
3.5x 10-3
Time [Sec]
Cha
nge
in F
C p
ower
gen
erat
ion
[pu]
GALMIPSOLMI
0 10 20 30-0.6
-0.5
-0.4
-0.3
-0.2
-0.1
0
0.1
0.2
Time [Sec]C
hang
e in
die
sel p
ower
gen
erat
ion
[pu]
PSOLMIGALMI
0 1 2 3 4 5-0.08
-0.06
-0.04
-0.02
0
0.02
Time [Sec]
Cha
nge
in B
ES
S p
ower
gen
erat
ion
[pu]
GALMIPSOLMI
0 2 4 6 8 10-0.5
0
0.5
1
1.5
2
2.5x 10-3
Time [Sec]
Cha
nge
in D
G p
ower
gen
erat
ion
[pu]
GALMIPSOLMI
0 1 2 3 4 5-2
-1
0
1
2
3
4
5x 10-3
Time [Sec]
Cha
nge
in p
ower
gen
erat
ion
in a
rea-
1 [p
u]
GALMIPSOLMI
0 2 4 6 8 10-2
0
2
4
6
8
10
12x 10-4
Time [Sec]
Cha
nge
in p
ower
gen
erat
ion
in a
rea-
2 [p
u]
GALMIPSOLMI
23
6.3 Effect of communication delays
The effect of communication time delay on the behaviour of the power system for GALMI controller is shown in
Fig. 15(a-c). Although the communication delay influences the controller performance, as discussed earlier, with PSOLMI the
controller performance is almost insensitive to the effect of communication delay, as shown in Fig. 15(d-f).
6.4 Robustness with parameter variations
The robustness of the controllers is evaluated by the computation of integral square error (ISE) for changes in parameters given
as:
302 2 2
1 20
( )T IEISE f f P dt (31)
The values of ISE are shown in Fig. 16. As observed for the change in parameters over a large percentage range with respect to
nominal value, the corresponding indices are higher in case of GALMI as compared to PSOLMI controller.
These results successfully validate that the proposed controller not only compensates against parameters variations but also
regulates the frequency deviation profile adequately.
24
(a) Frequency deviation in area-1 (b) Frequency deviation in area-2
(c)Tie-line power deviation (d) Frequency deviation in area-1
(e) Frequency deviation in area-2 (f) Tie-line power deviation
Fig. 15. Response for case study-2 with time delays
Fig. 16. Computed ISE for change in all system parameters
0 0.5 1 1.5 2 2.5 3-6
-4
-2
0
2x 10-3
Time [sec]
Freq
uenc
y de
viat
ion
F1 [p
u]
Delay time=1.5Delay time=3.5
0 0.5 1 1.5 2 2.5 3-2.5
-2
-1.5
-1
-0.5
0
0.5x 10-3
Time [sec]
Freq
uenc
y de
viat
ion
F2 [p
u]
Delay time=1.5delay time=3.5
0 0.5 1 1.5 2 2.5 3-15
-10
-5
0
5x 10-4
Time [sec]
Tie-
line
pow
er [p
u]
Delay time=3.5delay time=1.5
0 1 2 3-3
-2
-1
0
1x 10-3
Time [sec]
Freq
uenc
y de
viat
ion
F1 [p
u]
Delay time=3.5delay time=1.5
0 1 2 3-10
-5
0
5x 10-4
Time [Sec]
Freq
uenc
y de
viat
ion
F2 [p
u]
Delay time=3.5delay time=1.5
0 1 2 3-6
-4
-2
0
2x 10-4
Time [Sec]
Tie-
line
pow
er [p
u]
Delay time=3.5delay time=1.5
-30% -20% -10% 0% +10% +20% +30%0
1
2
3
4
5x 10-6
Parameter variation
Inte
gral
squ
are
erro
r [pu
]
PSOLMIGALMI
25
Thus, for the LFC problem, PSO is superior to GA due to its simplicity, better convergence and computational time. This is the
reason of improvement. Both GA and PSO are similar in the sense that these two techniques are population-based search methods
and they search for the optimal solution by updating generations. Since the two approaches are supposed to find a solution to a
given objective function but employ different strategies and computational effort, it is appropriate to compare their performance. It
should be noted that while the GA is inherently discrete, i.e. it encodes the design variables into bits of 0’s and 1’s, therefore it
easily handles discrete design variables, PSO is inherently continuous and must be modified to handle discrete design variables.
Compared with GA, PSO has some attractive characteristics. It has memory, so knowledge of good solutions is retained by all
particles; whereas in GA previous knowledge of the problem is destroyed once the population changes. PSO has constructive
cooperation between particles, so particles in the swarm share information between them.
7. Conclusions
In this paper, the LFC problem was presented using several control algorithms for the power system consisting of different energy
sources. The new contribution of this research was the successful simulation of the proposed PSOLMI controller, demonstrating
superior performance as compared to other controllers. The results illustrated superior control effort and guaranteed robust
performance as compared to H∞, LMI-LQR and GALMI controllers, against various uncertainties such as wind power variation
and load change. The PSOLMI controller effectiveness on minimizing frequency deviation was validated for a variation in the
parameters of about ±30% from nominal value. Hence, it was shown that PSOLMI controller has adequate disturbance rejection
properties and thereby robustness in its performance. This improves system reliability, minimizing grid frequency oscillation and
enhancing closed loop stability. Further, the comparison of the dynamics of case studies 1 and 2 suggested increased amplitude
and oscillation of frequency deviation profiles in case of existing thermal with DG resources that are interconnected to another
thermal system.
Acknowledgements
This work was supported by FEDER funds (European Union) through COMPETE and by Portuguese funds through FCT, under
Projects FCOMP-01-0124-FEDER-020282 (Ref. PTDC/EEA-EEL/118519/2010) and PEst-OE/EEI/LA0021/2013. Also, the
research leading to these results has received funding from the EU Seventh Framework Programme FP7/2007-2013 under grant
agreement no. 309048.
26
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